Exchange-Correlation Energies and
Correlation Holes for Some Two- and
Four-Electron Atoms along a Nonlinear
Adiabatic Connection in Density
Functional Theory
R. POLLET,1 F. COLONNA,1 T. LEININGER,2 H. STOLL,3
H.-J. WERNER,3 A. SAVIN1
1
Laboratoire de Chimie Théorique (CNRS et Université Pierre et Marie Curie), Paris, France
Laboratoire de Physique Quantique (CNRS et Université Paul Sabatier), Toulouse, France
3
Institut für Theoretische Chemie (Universität Stuttgart), Stuttgart, Germany
2
Received 28 February 2001; accepted 29 January 2002
DOI 10.1002/qua.10395
ABSTRACT: The adiabatic connection procedure of density functional theory has
been applied to two- and four-electron atomic systems by following a nonlinear path
that leads from the noninteracting Kohn–Sham reference system to the physical one. We
have calculated the exchange and correlation energies as the interaction strength is
increased, as well as the densities of the corresponding correlation holes. © 2002 Wiley
Periodicals, Inc. Int J Quantum Chem 91: 84 –93, 2003
Key words: correlation holes; density functional theory; exchange-correlation
energies; nonlinear adiabatic connection
Introduction
T
he adiabatic connection [1–3] offers the opportunity to understand and design approximate
functionals for the exchange-correlation energy of
density functional theory (DFT) [4]. The technique
Correspondence to: A. Savin; e-mail: andreas.savin@lct.
jussieu.fr
International Journal of Quantum Chemistry, Vol 91, 84 –93 (2003)
© 2002 Wiley Periodicals, Inc.
involves linking the Kohn–Sham [5] noninteracting
reference system to the fully interacting physical
system, by switching on the electron– electron interaction. The only necessary assumptions are the
adiabatic connectibility [6] and the choice of the
path to be followed at constant density. Among all
the possible adiabatic paths [7], a nonlinear one has
been tested in this article.
We briefly describe the underlying theory and
survey technical details. Then, beside the “exact”
EXCHANGE-CORRELATION ENERGIES AND CORRELATION HOLES
calculation of the exchange and correlation energies
of a few systems, we focus on the correlation holes
(see, e.g., Ref. [8]), because an understanding of
these can be a starting point for the elaboration of
new exchange-correlation functionals (see, e.g., the
Taylor expansion of Becke [9] or the real-space cutoff procedure of Perdew [10]).
Theory
In the conventional application of the adiabatic
connection procedure [11, 12], the electron– electron
interaction is switched on by using a linear interpolation, but alternative paths exist. For example,
we can make use of the error function and define
␭
⫽
V̂ ee
1
2
冘 erf共兩r␭兩r⫺ ⫺r 兩 r 兩兲
再
F ␭ 关 ␳ 兴 ⫽ max E␭关v␭兴 ⫺
v␭
冕
冎
␳共r兲v␭共r兲 dr
(3)
where
E ␭ 关v ␭ 兴 ⫽ 具⌽ v␭␭兩Ĥ ␭ 兩⌽ v␭␭典
(4)
If the arbitrary density ␳ is the physical one, we get
at the end of the maximization process the potential
␭
␭
v␭max and thus the ground-state wavefunction ⌽vmax
of the system in partial interaction, which yields the
density ␳.
␭
␭ , we can now calculate the
As we know ⌽vmax
corresponding exchange and correlation energies.
Because the density is constant, the ␭-dependent
correlation energy
N
i
i
i⫽j
j
(1)
␭
where the interaction strength ␭ lies between 0
(Kohn–Sham system) and ⬁ (physical system). Its
attractive features have already been discussed by
Yang [7]. Furthermore, it can give new insights into
a recent method that combines short-range density
functionals with long-range configuration interaction [13–17]. Adiabatic connection is indeed a rigorous formulation of such a coupling, and different
paths will obviously produce different short-range
exchange-correlation energies (at intermediate interaction strengths). One can then observe that approximations, for instance, LDA, can be more accurate for some paths [18].
We start by defining a ␭-dependent Hamiltonian,
␭
␭ ⫽0
␭ ⫽0
␭
␭
0
␭ 兩Ĥ 兩⌽ ␭ 典 ⫺ 具⌽ 0
E c␭ ⫽ 具⌽ v max
v max
v max 兩Ĥ 兩⌽ v max 典
j
(5)
becomes
␭
␭
␭ ⫽0
␭ ⫽0
␭
␭
0
␭ 兩T̂ ⫹ V̂ ee 兩⌽ ␭ 典 ⫺ 具⌽ 0
E c␭ ⫽ 具⌽ v max
v max
v max 兩T̂ ⫹ V̂ ee 兩⌽ v max 典
(6)
⫽ F ␭ 关 ␳ 兴 ⫺ F ␭ ⫽0 关 ␳ 兴 ⫺ 具⌽
␭ ⫽0
0
v max
␭
兩V̂ ee
兩⌽
␭ ⫽0
0
v max
典
(7)
For the ␭-dependent exchange energy, which depends only on the wavefunction of the noninteracting Kohn–Sham system, we get
␭ ⫽0
␭ ⫽0
0
0
E x␭ ⫽ 具⌽ v max
兩V̂ ee 共 ␭ 兲兩⌽ v max
典 ⫺ U␭
(8)
␭⫽0
冘 12 ⵜ ⫹ V̂ ⫹ 冘 v 共r 兲
N
Ĥ ␭ ⫽ ⫺
N
2
i
i
␭
ee
␭
i
(2)
i
where the first term is the kinetic energy operator
and the last term is the local potential associated
with the system with partial electron– electron interaction. When ␭ 3 ⬁, v␭3⬁ will be the real external potential of the interacting system, whereas
when ␭ ⫽ 0, v␭⫽0 is identical to the Kohn–Sham
potential. The potential v␭ has to be constructed so
that it maintains the density equal to the physical
one for all ␭’s. This can be achieved by searching for
the universal Legendre transform functional [19]
(see also [20]),
0
where ⌽vmax
typically is a single Slater determinant
of Kohn–Sham orbitals, and U␭ is the Hartree energy for the modified electron– electron interaction,
U ␭关 ␳ 兴 ⫽
1
2
冕冕
␳ 共r 1兲␳共r2兲erf共␭r12兲
dr1dr2
r12
(9)
␭
␭ , we can also deduce the ␭-dependent
From ⌽vmax
pair function [21],
P 2␭ 共r 1, r2兲 ⫽ N共N ⫺ 1兲
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
冕 冕
...
␭
␭
兩⌽vmax
⫻ 共x1, x2, . . . , xN兲兩2ds1 ds2 dx3 . . . dxN
(10)
85
POLLET ET AL.
where x represents both spatial and spin coordinates r and s. This pair function is the probability
density to find simultaneously two electrons at
points r1 in the volume element dr1 and r2 in dr2,
whatever their spin coordinates may be. The conditional probability is the density of the N ⫺ 1
remaining electrons at point r2 when one reference
electron is at point r1,
P
␭
cond
P 2␭ 共r 1, r2兲
共2兩1兲 ⫽
␳共r1兲
(11)
The exchange-correlation density hole at point r2 is
then defined by
␭
␭
␳ xc
共2兩1兲 ⫽ P cond
共2兩1兲 ⫺ ␳ 共r 2兲
(12)
This quantity is a measure of the influence of the
reference electron on the density of the remaining
electrons. With regard to these functions, we focus
on the correlation part of Eq. (12),
␭
␳ c␭ 共2兩1兲 ⫽ ␳ xc
共2兩1兲 ⫺ ␳ x 共2兩1兲
(13)
where ␳x is the density of the Fermi hole.
Technical Details
In practical applications, the method is composed of cycles in which a variation in the potential
is immediately followed by a wavefunction calculation. Thus, the technique consists in allowing a
“dialog” between a standard optimization program
and an ab initio software.
To optimize the local potential, a convenient
form was chosen, namely a spherical Gaussian basis,
v ps 共r兲 ⫽
冘 C r exp共⫺␥ r 兲 ⫹ Cr
i
pi
i
2
(14)
i
where r is the electron–nucleus distance, pi are integers greater than or equal to ⫺2, ␥i are positive
exponents and C may be deduced from asymptotical conditions. For example, the asymptotical behavior of the Kohn–Sham potential is given in Ref.
[22]
lim v␭⫽0共r兲 ⫽
r3⬁
86
⫺Z ⫹ N ⫺ 1
r
(15)
All these parameters are optimized by the simplex
method with the help of the amoeba procedure [23].
A typical plot of the maximization of the functional
(3) is represented in Figure 1 as an example.
A good criterion to judge the reliability of our
optimized parameters is to evaluate the quantity
used by Zhao and Parr [24],
⌬⫽
1
2
冕
关 ␳˜ ␭ 共r 1兲 ⫺ ␳共r1兲兴关␳˜ ␭共r2兲 ⫺ ␳共r2兲兴
dr1dr2
r12
(16)
where ␳˜ ␭ denotes the output electron density produced at the end of the maximizing process. Even if
accurate wavefunctions (hence, correlation energies) were obtained, the maximizing local potential
v␭ cannot be unambiguously determined. For example, a shift by a constant over the physically
relevant domain [25] or a rapidly oscillating perturbation [26] will not significantly affect the density.
As for the ab initio software, the Molpro [27]
program was modified by adding the new bielectronic integrals [16]. All the studied systems belong
to the isoelectronic series of helium and beryllium.
For He, C4⫹, and Ne8⫹, the reference ground state
density ␳ and the ground-state energy E␭ were obtained by full configuration interaction, whereas a
multireference configuration interaction [28, 29]
was used for Be and Ne6⫹. The basis functions are
uncontracted even tempered gaussians, up to f
functions (see Table I [11]).
Table II shows the accuracy of the maximizing
potentials by listing the minimum, maximum and
average values of the Parr criterion (16) for each
system.
Results and Discussion
EXCHANGE AND CORRELATION ENERGIES
The ␭-dependent exchange energies were calculated according to Eq. (8) for several interaction
strengths. Figures 2 and 3 show these results for
each series.
For the ␭-dependent correlation energies, we
performed nonlinear least-squares fitting of the
curves with a fit limit set to 1e-08, using Gnuplot
[30], according to the three-parameter fit function
E c␭ ⫽
␥ x 2 ⫹ E cx 4
␣ ⫹ ␤x2 ⫹ x4
(17)
VOL. 91, NO. 2
EXCHANGE-CORRELATION ENERGIES AND CORRELATION HOLES
FIGURE 1. Maximization of the functional (3) by the simplex method for the Ne6⫹ system at ␭ ⫽ 2 over a number
of iterations. When convergence is reached, ⌬ ⫽ 4.18.10⫺9 [Eq. (16)].
where x ⫽ ␭/Z. This function indeed satisfies
known exact conditions. The first ones are obviously
再
0 for ␭ ⫽ 0 共Kohn–Sham system兲
E c␭ ⫽ E
for ␭ 3 ⬁ 共physical system兲
c
(18)
dEc␭
1
⫽
d␭
冑␲
冕冕
␳共r1兲␳c␭共2兩1兲exp关⫺共␭r12兲2兴 dr1dr2 (19)
For ␭ ⫽ 0, the Gaussian function equals one and the
correlation hole integrates to zero so that
dEc␭
⫽0
d␭
Moreover, applying the Hellmann–Feynman theorem to Eq. (6) leads to
for ␭ ⫽ 0 共Kohn–Sham system兲
(20)
TABLE I ______________________________________________________________________________________________
Even-tempered Gaussian exponents obtained from rule ␣n ⴝ ␣c␦(2nⴚMⴚ1)/2.
Functions
s
p
d
f
Systems
M
␦
␣c
M
␦
␣c
M
␦
␣c
M
␦
␣c
He
C4⫹
Ne8⫹
Be
Ne6⫹
21
21
21
21
21
2,1
2,1
2,1
2,1
2,1
40,0
663,0
2030,0
4,0
130,0
7
7
7
9
9
2,1
2,1
2,1
2,1
2,1
3,0
34,0
100,0
1,0
15,0
5
5
5
5
5
2,1
2,1
2,1
2,1
2,1
3,0
34,0
100,0
1,0
15,0
3
3
3
3
3
2,1
2,1
2,1
2,1
2,1
3,0
34,0
100,0
1,0
15,0
M is the number of Gaussians, ␣c is the central exponent, and ␦ is the ratio between two consecutive exponents.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
87
POLLET ET AL.
TABLE II ______________________________________
Minimum (⌬min), maximum (⌬max) and average (⌬avg)
Parr criteria (16) along the adiabatic connection.
Systems
He
C4⫹
Ne8⫹
Be
Ne6⫹
⌬min
⌬max
⫺11
⌬avga
⫺7
2.15.10
3.02.10⫺13
1.72.10⫺14
2.56.10⫺9
1.47.10⫺9
⫺9
8.74.10
1.37.10⫺10
6.36.10⫺9
7.52.10⫺8
6.83.10⫺8
1.07.10
2.46.10⫺12
3.40.10⫺12
1.07.10⫺8
8.48.10⫺9
Equal to 10¥ilog(⌬i)/N, where N is the number of points along
the adiabatic connection.
a
When ␭ 3 ⬁, we use the definition of the Dirac
function,
␦ 共r 12 兲 ⫽ lim
␭3⬁
so that
再
冎
␭3
exp关⫺共␭r12兲2兴
␲3/ 2
(21)
dEc␭ ␲
⬇
d␭ ␭3
⫽
␲
␭3
⬇
␲
␭3
冕
冕
冕
␳共r1兲dr1
冕
␳c␭共2兩1兲␦共r12兲dr2
(22)
␳ 共r 1兲␳c␭共1兩1兲dr1
(23)
␳ 共r 1兲␳c共1兩1兲dr1
(24)
where we use a Taylor expansion of the correlation
hole near ␭ 3 ⬁ in Eq. (24). We thus obtain
dEc␭ ␲
⬀
d␭ ␭3
for ␭ 3 ⬁ 共physical system兲
(25)
In summary, the conditions (18), (20), and (25) are
all satisfied by the fit function (17).
Figures 4 and 5 show these results for each series. The resulting fit errors and parameters are
shown in Table III.
FIGURE 2. The ␭-dependent exchange energies for systems belonging to the helium series (He, C4⫹, and Ne8⫹).
88
VOL. 91, NO. 2
EXCHANGE-CORRELATION ENERGIES AND CORRELATION HOLES
FIGURE 3. The ␭-dependent exchange energies for systems belonging to the beryllium series (Be and Ne6⫹).
The asymptotical behavior (when ␭ 3 ⬁) of the
␭-dependent correlation energies is different between the two series. In the case of the helium
series, the three curves go to values that are close to
each other, whereas for the beryllium series, the
curves tend to values that are well separated. This
feature can be explained by writing the electronic
Hamiltonian in such a way that 1/r12 becomes a
perturbation with factor 1/Z [31]. One can thus
show that the second-order energetic correction is
proportional to a constant in the case of the helium
series, whereas the first-order energetic correction
is linear with Z for the beryllium series.
␳ c␭ 共2兩1兲 ⫽
(26)
where the electron density equals
␳ 共r兲 ⫽ 2兩␺共r兲兩2
(27)
where ␺ is the only occupied Kohn–Sham orbital,
␭
2
␭ 共r1, r2兲兩
P 2␭ 共r 1, r2兲 ⫽ 2兩⌽vmax
(28)
and where
␳ x共2兩1兲 ⫽ ⫺
CORRELATION HOLES
The Coulomb hole takes shape as the interaction
strength ␭ is increased and recovers its final shape
when ␭ 3 ⬁. This quantity is simple to obtain for
two-electron systems such as the helium series. According to Eq. (13), the ␭-dependent correlation
hole is
P 2␭ 共r 1, r2兲
⫺ ␳共r2兲 ⫺ ␳x共2兩1兲
␳共r1兲
␳共r2兲
2
(29)
is the Fermi hole, which corresponds to ␭ ⫽ 0, as in
a two-electron system exchange effects come only
from the self-interaction correction. Moreover, the
full CI wavefunction can be expanded in the natural
orbital basis [32],
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
89
POLLET ET AL.
FIGURE 4. Fits of the ␭-dependent correlation energies for systems belonging to the helium series (He, C4⫹, and
Ne8⫹) according to the fit function (17).
␭
␭ 共r 1, r2兲 ⫽
⌽ v max
冘 d ␨ 共r 兲␨ 共r 兲
i i
1
i
2
(30)
i
where d1 ⫽ 公␭1/2 and di ⫽ ⫺ 公␭i/2 for i ⬎ 1, ␭i
being the occupation number of the orbital ␨i.
Correlation holes for He have been studied for
two positions of the reference electron. For the first
series (when the reference electron is at 0.85 a.u.
from the nucleus), we give Figure 6, corresponding
to a weak interaction strength (for ␭ ⫽ 0.1, where
the amplitude equals 10⫺4 a.u.), as the shape basically remains unchanged when ␭ is increased (the
amplitude becomes respectively 0.045 a.u. for ␭ ⫽ 1,
0.14 a.u. for ␭ ⫽ 3, and 0.16 a.u. when ␭ 3 ⬁).
In this series, the shape of all holes results mainly
from angular correlation, which comes from mixing
with the 2p orbital. Whereas the probability to find
an electron with ␤ spin on the same side of the
nucleus as the reference electron with ␣ spin is
decreased, the probability of finding an electron
with ␤ spin on the opposite side is increased. The
90
amplitude of the 2px natural orbital is indeed close
to its maximum in this region (cf., e.g., Fig. 1 in Ref.
[33]). Thus, the excited configuration 2p2 is responsible for the angular correlation effects that are
observed.
For the second series (when the reference electron is at 0.4 a.u. from the nucleus), the initial
angular shape (for ␭ ⫽ 0.1, where the amplitude
equals 4.10⫺5 a.u.) is still observed (see Fig. 7).
But as ␭ is augmented, the shape of the hole is
then modified by radial correlation (see Fig. 8 for
␭ ⫽ 1, where the amplitude equals 0.07 a.u.), to
obtain a Coulomb hole where angular correlation is
very weak (the contour plot however exhibits an
asymmetrical shape).
For ␭’s greater than 1, the shape basically remains unchanged (the amplitude becomes 0.35 a.u.
for ␭ ⫽ 3 and 0.45 a.u. when ␭ 3 ⬁).
Thus, we observe that whatever the position of
the reference electron may be, the shape of the
correlation hole for weak interaction strengths is
VOL. 91, NO. 2
EXCHANGE-CORRELATION ENERGIES AND CORRELATION HOLES
FIGURE 5. Fits of the ␭-dependent correlation energies for systems belonging to the beryllium series (Be and Ne8⫹)
according to the fit function (17).
determined by angular correlation effects. To understand why, let us examine the third-order series
of our modified two-electron operator near ␭ ⫽ 0,
2␭3 2
erf共␭r12兲 2␭
⬇
⫺
r ⫹...
r12
冑␲ 3 冑␲ 12
(31)
We must then determine which bielectronic integrals contribute to the correlation energy. If the
TABLE III _____________________________________
Fitting parameters and root mean square (RMS) of
the ␭-dependent correlation energies according to
the fit function (17).
Systems
He
C4⫹
Ne8⫹
Be
Ne6⫹
RMS
␣
␤
␥
0.000107
0.000226
0.000101
0.001253
0.003603
0.11118
0.34688
0.24156
0.05138
0.28233
1.23083
1.53225
1.73407
1.58206
3.73581
0.00140
0.00093
0.00197
⫺0.07969
⫺0.58459
only virtual orbitals are 2s and 2p, the integrals are
(1s2s兩1s2s) and (1s2p兩1s2p) (in chemists’ notation).
The first term in the series is a constant, so that the
corresponding integrals vanish because the atomic
orbitals are orthonormal. The second term depends
on the bielectronic operator r212 ⫽ r21 ⫹ r22 ⫺ 2r1 䡠 r2.
The first two operators are monoelectronic so that
both integrals still vanish. Only the last operator
yields a nonzero contribution to the correlation energy. It is a product of two monoelectronic operators so that we can split the two integrals. For
symmetry reasons, the only nonzero integrals are of
the type 兰 1s(ri)xi2px(ri)d ri. It is this peculiar type of
integral that is responsible for all the angular effects
when the interaction strength is weak, and no radial
correlation effects exist.
Conclusion
Among all possible adiabatic connection paths
that link the noninteracting reference system to the
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
91
POLLET ET AL.
FIGURE 6. Correlation hole of He for ␭ ⫽ 0.1 when the reference electron is at position x ⫽ 0.85 a.u. and y ⫽ 0.0
a.u. On the surface plot, the N label represents the nucleus position, and the e label, the reference electron position.
physical one, a nonlinear one that involves the error
function was studied. By searching for the universal functional defined by Lieb, that is by optimizing
the local potential that is mapped with the physical
electron density, we have calculated the exchange
and correlation energies along that adiabatic con-
nection path for systems that belong to the helium
and beryllium isoelectronic series. We have also
computed the correlation holes of helium as the
interaction strength increases and observed that angular correlation effects always prevail when this
strength is weak. Such data should serve for the
FIGURE 7. Correlation hole of He for ␭ ⫽ 0.1 when the reference electron is at position x ⫽ 0.4 a.u. and y ⫽ 0.0
a.u. On the surface plot, the N label represents the nucleus position, and the e label, the reference electron position.
92
VOL. 91, NO. 2
EXCHANGE-CORRELATION ENERGIES AND CORRELATION HOLES
FIGURE 8. Correlation hole of He for ␭ ⫽ 1.0 when the reference electron is at position x ⫽ 0.4 a.u. and y ⫽ 0.0
a.u. On the surface plot, the N label represents the nucleus position, and the e label, the reference electron position.
improvement of current density functionals or for
the design of new approximations.
17. Pollet, R.; Savin, A.; Leininger, T.; Stoll, H. J Chem Phys,
2002, 116, 1250.
18. Savin, A., Colonna, F.; Pollet, F., Int J Quantum Chem, in
press.
References
19. Lieb, E. Int J Quantum Chem 1983, 24, 243.
20. Nalewajski, R. F.; Parr, R. G. J Chem Phys 1982, 77, 399.
1. Harris, J.; Jones, R. O. J Phys F 1974, 4, 1170.
2. Langreth, D. C.; Perdew, J. P. Solid State Commun 1975, 17,
1425.
3. Gunnarsson, O.; Lundqvist, B. I. Phys Rev B 1976, 13, 4274.
4. Hohenberg, P.; Kohn, W. Phys Rev B 1964, 136, 864.
5. Kohn, W.; Sham, L. J. Phys Rev A 1965, 140, 1133.
6. Harris, J. Phys Rev A 1984, 29, 1648.
7. Yang, W. J Chem Phys 1998, 109, 10107.
8. Buijse, M. A.; Baerends, E. J. Density Functional Theory of
Molecules, Clusters, and Solids; Kluwer Academic Publishers: Netherlands, 1995. p. 1.
9. Becke, A. D. Int J Quantum Chem 1983, 23, 1915.
10. Perdew, J. P. Phys Rev Lett 1985, 55, 1665.
11. Colonna, F.; Savin, A. J Chem Phys 1999, 110, 2828.
12. Joubert, D. P.; Srivastava, G. P. J Chem Phys 1998, 109, 5212.
13. Stoll, H.; Savin, A. Density Functional Methods in Physics;
Plenum: New York, 1985. p. 327.
14. Savin, A.; Flad, H.-J. Int J Quantum Chem 1995, 56, 327.
15. Savin, A. Recent Developments and Applications of Modern
Density Functional Theory; Elsevier: Amsterdam, 1996. p.
327.
16. Leininger, T.; Stoll, H.; Werner, H.-J.; Savin, A. Chem Phys
Lett 1997, 275, 151.
21. McWeeny, R.; Sutcliffe, T. Methods of Molecular Quantum
Mechanics; Academic Press: London, 1976.
22. Levy, M.; Perdew, J. P.; Sahni, V. Phys Rev A 1984, 30, 2745.
23. Press, W.; Teukolsky, S.; Vetterling, W.; Flannery, B. Numerical Recipes; Cambridge University Press: Cambridge, 1992.
24. Zhao, Q.; Parr, R. G. Phys Rev A 1992, 46, 2337.
25. Levy, M. Private communication.
26. Kohn, W. Private communication.
27. Molpro is a package of ab initio programs written by
Werner, H.-J.; Knowles, P. J.; with contributions from Amos,
R. D.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn,
A. J.; Eckert, F.; Hampel, C.; Hetzer, G.; Leininger, T.; Lindh,
R.; Lloyd, A. W.; Meyer, W.; Mura, M. E.; Nicklaß, A.;
Palmieri, P.; Peterson, K.; Pitzer, R.; Pulay, P.; Rauhut, G.;
Schütz, M.; Stoll, H.; Stone, A. J.; Thorsteinsson, T.
28. Werner, H.-J.; Knowles, P. J. J Chem Phys 1988, 89, 5803.
29. Knowles, P. J.; Werner, H.-J. Chem Phys Lett 1985, 145, 514.
30. Williams, T.; Kelley, C.; et al. Gnuplot Linux version 3.7,
Copyright 1986 –1993, 1998, 1999.
31. Linderberg, J.; Shull, H. J Mol Spectr 1960, 5, 1.
32. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry;
McGraw-Hill: New York, 1989.
33. Davidson, E. R. Rev Mod Phys 1972, 44, 451.
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